Novel asymptotic methods for wave propagation

Thesis (Ph. D.)--University of Rochester. Institute of Optics, 1997.Asymptotic techniques give accurate approximations in the modelling of wave propagation when the wavelength is much smaller than any other length scale in the system. These methods rely principally on ray information. The optical properties of an inhomogeneous medium can be characterized by a function called the wave propagator, and the derivation of an asymptotic expression for this function is given in this thesis. The limitations to the validity of the resulting expression are also treated here and they relate to the presence of caustics. The central contribution, however, is the development of a collection of new tools for dealing with the difficulties associated with caustics. The fractional Legendre transformation (FLT) is among these tools, and gives a continuous transition between a function and its Legendre transform. Also proposed is a generalization of Hamilton's formalism for geometrical optics. This generalization allows for the definition of countless new representations for the characteristic functions that characterize the system in the ray domain. A subset of these new representations, associated with the FLT, is shown to have the fractional Fourier transform of a field as its counterpart in the wave domain. The asymptotic relation between these representations is the key to the derivation of a new asymptotic form for the wave propagator that has no difficulties with caustics and automatically accounts for the caustic phase shifts